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Pricing and Hedging withoutFast Fourier Transform
A revisited, reliable and stable Fourier Transform Method for Affine Jump Diffusion Models
Marcello Minenna - Paolo VerzellaStructured Products Europe – Nov 3, 2005 - London
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Syllabus of the presentation
• Review of Fourier Methods in Option Pricing
• Calibration and Performance
• Greek Behavior of New FT-Q
3
Review of Fourier Methods in Option Pricing – theory
European Call Maturity Terminal Spot Price
In AJD models Call Price can be expressed in a form close to the canonical Black – Scholes - Merton style
where
under different martingale measures
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determined by using the Levy’s inversion formula, i.e.:
can be
under different martingale measures
Review of Fourier Methods in Option Pricing – theory
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requires
a close formula for the Characteristic Function of the log – terminal price, i.e.:
Review of Fourier Methods in Option Pricing – theory
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has
a closed formula for AJD models
Review of Fourier Methods in Option Pricing – theory
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How to compute:
Quadrature AlgorithmFT – Q for Fast Fourier Transform
FFT
Review of Fourier Methods in Option Pricing – practice
I II
a b
Old FT - Q New FT - Q
tC
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Review of Fourier Methods in Option Pricing – practice
Algorithms Valuation Criteria
The algorithm is defined stable if and only if
STABILITY
it "closes" the quadrature scheme
a "reasonable" result different from
a NaN value
spanning
the pricing formula on a vast area of
the parameters set.gives
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Review of Fourier Methods in Option Pricing – practice
Algorithms Valuation Criteria
Call via standard Black –Scholes – Merton
ACCURACYThe algorithm is defined accurate if and only if
pricing pricing
3up to 10 precision-
the Call under the Black -Scholes - Merton model via:
Old FT - Q
New FT - Q
FFT
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Review of Fourier Methods in Option Pricing – practice
Algorithms Valuation Criteria
The algorithm is defined fast with respect to
SPEED
the results of the FFT algorithm
a set of 4100 prices along the strike
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High Order Newton Cotes Algorithm
Review of Fourier Methods in Option Pricing – practice
Quadrature AlgorithmOld FT - Q
through
Up to 8th
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Pros (+)
STABILITYSPEED
Review of Fourier Methods in Option Pricing – practice
Quadrature AlgorithmOld FT - Q
through
Cons (-)
ACCURACY
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Review of Fourier Methods in Option Pricing – practice
Quadrature AlgorithmNew FT - Q
through
In order to overcome the cited problems of Old FT – Q:
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Review of Fourier Methods in Option Pricing – practice
Quadrature AlgorithmNew FT - Q
through
In order to overcome the cited problems of Old FT – Q:
Gauss - Lobatto Quadrature Algorithm
Re-adjustment of
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Pros (+)
STABILITY SPEED
Review of Fourier Methods in Option Pricing – practice
through
Cons (-)
ACCURACY
Quadrature AlgorithmNew FT - Q
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Cooley - Tukey algorithm
Review of Fourier Methods in Option Pricing – practice
throughFast Fourier Trasform
FFT
Applied to the equivalent formula via a recombinant FFT parameters
for ATMln
ln
0( )
Ki K
t jeC e f d
af f f
p
- ¥-= ò
tC
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Pros (+)
STABILITYSPEED
Review of Fourier Methods in Option Pricing – practice
through
Cons (-)
Fast Fourier TrasformFFT
ACCURACY
FASTER(up to 20 times the quadrature algorithms)
* The formula must be changed arbitrarilyaccording to Option moneyness
** the recombinant FFT parameters must be changed according to the choice of the
pricing models
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Syllabus of the presentation
• Review of Fourier Methods in Option Pricing
• Calibration and Performance
• Greek Behaviour of New FT-Q
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The Calibration Procedure and Performance
Quadrature AlgorithmFT - Q
Fast Fourier TrasformFFT
I II
a b
Old FT - Q New FT - Q
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through Quadrature AlgorithmOld FT - Q
Pros (+)
STABILITY
SPEED
Cons (-)
ACCURACY
The Calibration Procedure and Performance
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through
Pros (+)
SPEED STABILITY *
Cons (-)
ACCURACY **
Fast Fourier TrasformFFT
The Calibration Procedure and Performance
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through Quadrature AlgorithmNew FT - Q
Pros (+)
STABILITY
SPEED
Cons (-)
ACCURACY
The Calibration Procedure and Performance
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By keeping in mind that only New FT-Q is stable and accurate, some figures on speed
Original Option Pricing Formulas are used
FFTHeston Model Merton Model BCC Model
7.26 sec. 10.54 sec. 18.33 sec.
NEW FT - QHeston Model Merton Model BCC Model
55.12 sec. 66.48 sec. 110.39 sec.
OLD FT – QHeston Model Merton Model BCC Model
390.41 sec. 454.76 sec. 722.1 sec.
By now, the speed of Fourier Trasform method is closer than ever to the FFT calibration time
The Calibration Procedure and Performance
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FFTHeston Model Merton Model BCC Model
7.24 sec. 10.54 sec. 18.32 sec.
NEW FT - QHeston Model Merton Model BCC Model
23.13 sec. 66.48 sec. 48.7 sec.
OLD FT – QHeston Model Merton Model BCC Model
331.6 sec. 454.76 sec. 688.5 sec.
Calibration Performances using Option Readjusted Pricing Formulas
where available
The Calibration Procedure and Performance
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Syllabus of the presentation
• Review of Fourier Methods in Option Pricing
• Calibration Procedure and Performance
• Greek Behaviour of New FT-Q
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An impressive methodology to test Stability of theNew FT – Quadrature algorithm is to compute Greeks
Infact, in an AJD setting the Greeks are available in closed form
So, an extended spanning of the AJD Greeks on the parameters set is useful to assess models and test
Stability
Greek behaviour of new FT-Q
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
v S
Black – Scholes Delta
0
0.9
200
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Lambda = -2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
0.9
v 0 S 200
Heston Delta
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2000
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0.9
v 0
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
Heston Delta Lambda = 2
S 200
30
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0.9
v
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Lambda = 0
200
Heston Delta Rho = -1
0 S
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0.9
0v
Heston Delta
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Lambda = 0
Rho = 1
200S
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0
0,2
0,4
0,6
0,8
1
1,2
EtaJ = 0
MuJ= 0
0.9
v0
LambdaJ = 0Merton Delta
S 200
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
EtaJ = 0.1
MuJ= 0.5
0.9
v0
Merton Delta LambdaJ = 1.8
200S
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Merton Delta EtaJ = 0.1
LambdaJ = 0.5
MuJ= 0.5
0.9
v0 S 200
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v0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
LambdaJ = 0.5
MuJ= 0.5
Merton Delta EtaJ = 0.75
0.9
0 S 200
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
LambdaJ = 0.5
Eta= 0.1
Merton Delta MuJ = 2.5
0.9
v0 S 200
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vS0
0.9
2000
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Black – Scholes Gamma
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0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Heston GammaLambda = -2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
200S0v
0.9
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0
0,002
0,004
0,006
0,008
0,01
0,012
0,014
0,016
0,018
Heston GammaLambda = -2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
200S0v
0.9
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0
0,005
0,01
0,015
0,02
0,025
Lambda = 2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
Heston Gamma
0 S 200v
0.9
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0
0,002
0,004
0,006
0,008
0,01
0,012
0,014
0,016
0,018
0,02
Heston Gamma Rho = -1
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Lambda = 0
0.9
v 0 S 200
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0
0,002
0,004
0,006
0,008
0,01
0,012
0,014
0,016
0,018
Heston Gamma Rho = 1
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Lambda = 0
0.9
v0 S 200
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0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Merton Gamma LambdaJ = 0
EtaJ = 0
MuJ= 0
0.9
v 0 S 200
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0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
v
0.9
200S
EtaJ = 0.1
MuJ = 0.5
Merton Gamma LambdaJ = 0.9
0
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0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Merton Gamma LambdaJ = 1.8
EtaJ = 0.1
MuJ = 0.5
0.9
v S 2000
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0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
0,04
0,045
0,05
Merton Gamma LambdaJ = 1.8
EtaJ = 0.1
MuJ= 0.5
0.9
v 0 S 200
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0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
LambdaJ = 0.5
MuJ= 0.5
0.9
v 0 S 200
Merton Gamma EtaJ = 0.1
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0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Merton Gamma EtaJ = 0.75
LambdaJ = 0.5
MuJ= 0.5
0.9
v 0 S 200
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0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
Merton Gamma MuJ = 2.5
LambdaJ = 0.5
EtaJ= 0.1
0.9
v0 S 200
50
0
5
10
15
20
25
30
v
0.9
2000 S
Black – Scholes Vega
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0
50
100
150
200
250
Heston VegaLambda = -2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
0.9
v 0 200S
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0
5
10
15
20
25
30
35
40
Heston VegaLambda = -2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
S 200
0.9
v 0
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0
5
10
15
20
25
30
35
40
Heston Vega Lambda = 2
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Rho = 0
0.9
v 0 S 200
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0
5
10
15
20
25
30
35
40
Heston Vega Rho = -1
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Lambda = 0
0.9
v 0 S 200
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0
5
10
15
20
25
30
35
40
Heston Vega Rho = 1
CappaV = 2
ThetaV = 0.3
EtaV = 0.1
Lambda = 00.9
v
0 S 200
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0
50
100
150
200
250
300
350
400
450
500
Merton Vega LambdaJ = 0
EtaJ = 0
MuJ= 0
0.9
v 0 S 200
57
0
50
100
150
200
250
300
350
400
450
500
EtaJ = 0.1
MuJ = 0.5
LambdaJ = 0.9Merton Vega
200S0v
0.9
58
0
50
100
150
200
250
300
350
400
450
500
Merton Vega LambdaJ = 1.8
EtaJ = 0.1
MuJ= 0.5
0.9
v 0 S 200
59
0
50
100
150
200
250
300
350
400
450
500
Merton Vega EtaJ = 0.1
LambdaJ = 0.5
MuJ= 0.5
0.9
v 0 S 200
60
0
50
100
150
200
250
300
350
400
450
500
Merton Vega EtaJ = 0.75
LambdaJ = 0.5
MuJ= 0.5
0.9
v 0 S 200
61
0
50
100
150
200
250
300
350
400
450
500
Merton Vega MuJ = 2.5
LambdaJ = 0.5
EtaJ= 0.5
0.9
v 0 S 200
